#273039

order	512
nilpotency class	3
isoclinism	isoclinic class of #272923
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#538	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1097 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 36 points:

(5,6)(7,13,15,11)(8,10)(9,16,17,12)(14,18)(19,20)(21,22)(23,31)(24,28)(25,34)(26,30)(27,32)(29,35)(33,36);(1,2)(3,4)(5,7,10,15)(6,11,8,13)(9,14,17,20)(12,18,16,19)(21,23,26,24)(22,27,30,36)(25,33,35,32)(28,34,31,29);(2,4)(5,8,10,6)(7,13,15,11)(9,16,17,12)(14,19,20,18)(21,24,26,23)(22,28,30,31)(25,33,35,32)(27,34,36,29);(5,9)(6,12)(7,14)(8,16)(10,17)(11,18)(13,19)(15,20)(22,29)(25,35)(27,28)(30,34)(31,36)(32,33)

Polycyclic presentation:

2^9|a^2=fhi,b^2=h,c^2=h,e^2=i,b^a=be,c^a=cf,d^a=dei,e^a=eh,c^b=cg,e^b=ei,f^b=fhi,e^c=ehi,e^d=ei,f^d=fhi

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 180235);